Nondestructive optical techniques for simultaneously measuring optical constants and thicknesses of single and multilayer films

ABSTRACT

An optical technique (apparatus and method based on the use of power spectral density analysis of spectroscopic multiple angle reflection and transmission data is disclosed. The apparatus and methods measure optical constants (n, k) and thicknesses of single and multilayer films. The apparatus and method provide for index determination with high accuracy (0.00001).

BACKGROUND

The present invention relates generally to optical techniques formeasuring the thickness of optical films, and more particularly, toapparatus and methods using nondestructive optical techniques forsimultaneously measuring optical constants and thicknesses of single andmultilayer films.

Currently, there are several methods to determine the optical constantsof thin films. These methods are described in the following references.The "Handbook of Optical Constants of Solids", edited by E. D. Palik(academic Press, N.Y., 1985) describes measurement of normal-incidencereflectance and transmittance over a wide spectral range.

The "Handbook of Optical Constants of Solids II", edited by E. D. Palik(academic Press, N.Y., 1991) describes measurement of R and T for normaland oblique angles of incidence (45°; 60°) for the polarizations TE andTm, over a wide spectral range. In "Physics of Thin Films", Vol. 2, byO. S. Heavens, edited by G. Hass and R. E. Thum (Academic Press, N.Y.,1964), using ellipsometry to measurement the polarization states ofcollimated monochromatic light before and after reflection from asurface to obtain the ratio r=R_(p) /R_(s),=tanψ exp(i Δ) of the complexp and s reflection coefficients is discussed. These methods ofdetermining the optical constants (n and k), however, are complicatedand at times yield inaccurate results.

The major disadvantage of the first two methods is that the opticalconstants are determined from the magnitude of the reflection spectrumwhich cannot be measured accurately (ΔR˜±0.3%). The measurement errorsin the reflection magnitude introduce significant errors in the opticalconstants (Δn˜±1%).

In general, the three above-cited methods are not appropriate fordetermining simultaneously and accurately the optical constants andthickness of relatively thick layers (greater than 20λ; greater than 10μm, where λ is the wavelength).

It would therefore be desirable to have an optical film thicknessmeasuring apparatus and method that overcomes the limitations ofconventional approaches. Accordingly, it is an objective of the presentinvention to provide for nondestructive optical techniques, includingapparatus and methods, for simultaneously measuring optical constantsand thicknesses of single and multilayer films.

SUMMARY OF THE INVENTION

To accomplish the above and other objectives, the present inventionovercomes the difficulties of prior art approaches by using the conceptof relative shift (ratio) of power spectral density as a function ofincident angle to simultaneously measure optical constants and thicknessof single and multilayer films. The present method is particularlyapplicable to thick films of greater than λ4.

The present invention measures reflectance, transmittance, or polarizedreflectance or transmittance for normal and oblique angles of incidenceover a wide spectral range, using a spectrophotometer, for example. Thefollowing methodology is then used to determine the optical constants(n, k) and thickness of single and multilayer films.

Power spectral density functions of the measured spectra (i.e.,reflectance, transmittance, or polarized reflectance or transmittance)for normal and oblique angles of incidence as a function of frequencyare calculated. Statistically significant peaks of the power spectraldensity functions are then determined. For a single layer only the mostsignificant peak (maximum peak) is considered. The positions of thepeaks of power spectral density functions at angles of incidence θ_(k),where θ₀ represent normal incidence are determined. The unknownparameters of the film are modeled using a dispersion formula that iscapable of describing the dispersion in the optical constants (n, k) inthe measured wavelength range. The dispersion formula is used to producesimulated values for the locations of the statistically significantpeaks and the ratio of the simulated locations of the peaks.

Then, the thicknesses and coefficients of the dispersion formula(optical constants) of the unknown layers are determined. Nonlinearglobal optimization algorithms are used to minimize an error function(merit function or loss function) of experimental (measured) andtheoretical data (modeled). Depending on the optimization technique useda single merit (error) function may be defined for all the datarespectively. For gradient optimization methods each fitted parameterhas its own merit function which is minimized/maximized.

Each set of data can have different weighting factors. The presentinvention is not sensitive to errors in the magnitude of the measuredreflectance, transmittance, or polarized reflectance or transmittance.In the extreme case of large errors in the magnitude of the reflectance,transmittance, or polarized reflectance or transmittance, one can fitonly on the power spectral density parameters. A consistency check maybe achieved by repeating the above steps at multiple angles anddifferent polarization.

BRIEF DESCRIPTION OF THE DRAWINGS

The various features and advantages of the present invention may be morereadily understood with reference to the following detailed descriptiontaken in conjunction with the accompanying drawings, wherein likereference numerals designate like structural elements, and in which:

FIG. 1 illustrates instrument configurations in which a simultaneousspectroscopic measurement at multiple angles of the surface of a film orwafer is performed in accordance with the principles of the presentinvention;

FIG. 2 is a flow chart of a measurement and calculation method inaccordance with the principles of the present invention;

FIG. 3 is a flow chart that details the calculation method used in themethod of FIG. 2;

FIG. 4 illustrate input variables used to define the error function(penalty function) used in calculation method of FIG. 3;

FIG. 5 illustrates simulated reflection as a function of wavelength fortwo example files with optical thickness of 20 μm and constant index ofrefraction of 2 and 1.5, respectively, and wherein the data wassimulated at normal incidence (0 degrees);

FIG. 6 illustrates simulated reflection as a function of wavelength fortwo example film with optical thickness of 20 μm and constant index ofrefraction of 2 and 1.5, respectively, and wherein the data wassimulated at 70 degrees;

FIG. 7 illustrates simulated power spectral density for two examplefilms with optical thickness (NT) of 20 μm and constant index ofrefraction of 2 and 1.5, respectively, and wherein the data wassimulated at both 0 and 70 degrees;

FIG. 8 illustrates simulated power spectral density for two examplefilms with fixed thickness of 10 μm and constant index of refraction of2 and 1.5, respectively, and wherein the data was simulated at both 0and 70 degrees;

FIG. 9 illustrates reflection amplitude as a function of wavelength ofpolyimide on silicon dioxide on silicon, and wherein the data wassimulated at both 0 and 70 degrees; and

FIG. 10 illustrates power spectral density of polyimide (8 μm) onsilicon dioxide (2 μm) on silicon, and wherein the data was simulated atboth 0 and 70 degrees.

DETAILED DESCRIPTION

Referring to the drawing figures, FIG. 1 illustrates apparatus 10 inwhich a simultaneous spectroscopic measurement at multiple angles of thesurface of a film 13 (or wafer 13) is performed using a measurement andcalculation method 20 in accordance with the principles of the presentinvention. The apparatus 10 comprises a first light source 11 and afirst detector 12 that are used to make spectroscopic measurements at anormal angle relative to the (single or multilayer) film 13 formed on asubstrate 14. A second light source 15 and a second detector 16 are usedto make spectroscopic measurements at an oblique angle of incidencerelative to the film 13. A controller 17 is used to control each of thelight sources 11, 15 and detectors 12, 16. Outputs of the respectivedetectors 12, 16 are processed by a computer 18.

These instruments and components are processed using a measurement andcalculation method 20 in accordance with the principles of the presentinvention. The measurement and calculation method 20 produces an output19 comprising the thickness of the (single or multilayer) film 13,optical constants n(λ) and k(λ) of the film 13, surface roughness data,composition data, and energy band gap data for the film 13. The presentinvention will be described in more detail with reference to FIG. 2,which is a flow chart illustrating the measurement and calculationmethod 20 in accordance with the principles of the present invention.

Referring to FIG. 2, it is a flow chart of one embodiment of themeasurement and calculation method 20 in accordance with the principlesof the present invention. In practicing the present method 20, thereflection spectra from the film 13 is measured 21 at normal incidence(angle=θ₀) using the first light source 11 and first detector 12. Thepower spectral density of the reflection spectra at θ₀ is computed 22.The reflection spectra from the film 13 is also measured 23 at theoblique angle of incidence (angle=θ_(k)) using the second light source15 and second detector 16. The power spectral density of the reflectionspectra at θ_(k) is computed 24. Statistically significant peaks (ξ) ofthe power spectral density spectra are then determined 25. The unknownoptical constants are modeled 26. Then, the power spectral densitymeasurements, the statistically significant peaks of the power spectraldensity spectra and the modeled optical constants are processed(compared 27) to determine 27 or calculate 27 the unknown thicknessesand coefficients of the dispersion formula (optical constants) of thefilm 13.

FIG. 3 is a flow chart that details the calculation method used themeasurement and calculation method 20 shown in FIG. 2. The measured data21, 23 are input to an error function 31 that is to be minimized. Themodeled 26 unknown optical constants are input to a general multilayersimulator 32. FIG. 4 illustrates input variables used to define theerror function 31 (penalty function) used in the calculation method ofFIG. 3. The input variables are those that are modeled 26.

The output of the general multilayer simulator 32 is also input to theerror function 31 that is to be minimized. The output of the errorfunction 31 that is to be minimized is input to an optimization process33 that uses a nonlinear global optimization algorithms to minimize amerit function (loss function or error function) of the experimental(measured) and theoretical data (modeled) data. The optimization process33 is iterated 34 until a solution is found. Once a solution is found,the optimization process 33 produces the outputs 19 of the method 20.

The present invention uses the concept of relative shift (ratio) ofpower spectral density as a function of incident angle to simultaneouslymeasure optical constants and thickness of single and multilayer films13. The present method is particularly applicable to thick films 13 ofgreater than λ/4. The present invention will be described below in moredetail, in terms of a mathematical description.

The method 20 measures reflectance (or transmittance) for normal andoblique angles of incidence (for example 45°; 70°) over a wide spectralrange, using a spectrophotometer (the detectors 12, 16), for example.The method 20 then determines the optical constants (n, k) and thicknessof single and multilayer films 13.

More specifically, in the present method 20, the power spectral densityof the measured 21, 23 reflection spectra for normal and oblique anglesof incidence as a function of frequency are calculated 22, 24. Thestatistically significant peaks of the power spectral density aredetermined 25. For a single layer one needs to consider only the mostsignificant peak (maximum peak). The parameter ξ^(j).sub.θ.sbsb.k isdefined as the position of peak j of power spectral density at angle ofincidence θ_(k), where θ₀ represents normal incidence. Morespecifically, the parameters ξ^(j).sub.θ.sbsb.0, ξ^(j).sub.θ.sbsb.1 andξ^(j).sub.θ.sbsb.1 /ξ^(j).sub.θ.sbsb.0 are determined 25.

The unknown optical constants are modeled 26 using a general dispersionformula that is capable of describing the dispersion in the opticalconstants (n, k) in the measured wavelength range. The generaldispersion formula may be one discussed in "Parameterization of theoptical functions of amorphous materials in the interband region", by G.E. Jelison, Jr. and F. A. Modine, Appl. Phys. Lett. 69 (3), 15 Jul.1996, and Appl. Phys. Lett. 69 (14), 30 Sep. 1996, and in the "FilmTekManual" by Scientific Computing International, 1998.

The general dispersion formula developed by the present inventor is ageneralization of the Lorentz oscillator model which defines a complexdielectric function ε=ε₁ +Iε₂, wherein ε₁ comprises a real part and Iε₂comprises an imaginary part, and is defined as follows: ##EQU1## whereε₁ and ε₂ are the real and imaginary part of the dielectric function,ε.sub.∞ is the high-frequency lattice dielectric constant,(E_(center))_(j) is the center energy of each oscillator. The physicalsignificance of the center energy depends on the material type andspectral range being considered. For example, in modeling semiconductormaterials the center energy is related to the transverse phononfrequency. A_(j) is the amplitude (strength) of each oscillator. Inmodeling semiconductor materials the Amplitude is related to both thetransverse and longitudinal phonon frequencies (Amplitude=√ω_(L) ²-ω_(T) ² ). ν_(j) is the vibration frequency (broadening) of the "j"oscillator. E is the energy and α is the damping coefficient. In thelimit of α=0 the dispersion formula reduces to the Lorentz oscillatormodel. The above equation is quite general and accurately applies tosemiconductor, dielectric, amorphous, crystalline and metallicmaterials.

Then, thicknesses and coefficients of the dispersion formula (opticalconstants) of the unknown layers are determined 27. Nonlinear globaloptimization algorithms are used to minimize a merit function (lossfunction or error function) of experimental (measured) and theoreticaldata (modeled) data. The nonlinear global optimization algorithms areused to minimize the merit function of:

ξ^(j).sub.θ.sbsb.0, ξ^(j).sub.θ.sbsb.1 and ξ^(j).sub.θ.sbsb.1/ξ^(j).sub.θ.sbsb.0, or

the power spectral density spectrums, or

the reflection spectra for normal and oblique angles of incidence.

Depending on the optimization technique used a single merit function(error function) is defined for all data, namely: ##EQU2## whereY_(expj) and Y_(calculatedj) represent the experimental and calculated(simulated) data of parameter j. β=1,2 for Absolute deviation and leastsquare options respectively. On the other hand, for gradientoptimization methods each fitted parameter has its own merit functionwhich is minimized/maximized, namely:

    (Merit Function).sub.j ={|Y.sub.targetj -Y.sub.calculatedj |.sup.β ×weight.sub.j }.sup.1/β

Each set of data can have different weighting factors. It should beunderstood that the present invention is not sensitive to errors in themagnitude of the measured reflection. In the extreme case of largeerrors in the magnitude of the reflection one can fit only on the powerspectral density parameters.

A consistency check may be achieved by repeating the above method stepsat multiple angles and different polarization.

Presented below are two examples illustrating use of the present method20.

EXAMPLE 1

To illustrate the significance of the present method 20, consider asimple example case of a single transparent film layer (k=0) withconstant index of refraction (no dispersion). Reference is made to FIGS.5-8 which pertain to Example 1.

FIG. 5 illustrates simulated reflection as a function of wavelength fortwo example films 13 with optical thickness of 20 μm and constant indexof refraction of 2 and 1.5, respectively, and wherein the data wassimulated at normal incidence (0 degrees). FIG. 6 illustrates simulatedreflection as a function of wavelength for two example films 13 withoptical thickness of 20 μm and constant index of refraction of 2 and1.5, respectively, and wherein the data was simulated at 70 degrees.FIG. 7 illustrates simulated power spectral density for two examplefilms 13 with optical thickness (NT) of 20 μm and constant index ofrefraction of 2 and 1.5, respectively, and wherein the data wassimulated at both 0 and 70 degrees. FIG. 8 illustrates simulated powerspectral density for two example films 13 with fixed thickness of 10 μmand constant index of refraction of 2 and 1.5, respectively, and whereinthe data was simulated at both 0 and 70 degrees.

If one performs a power spectral density measurement, PSD(R₀), powerspectral density analysis of the reflection magnitude at normalincidence (FIG. 5) as a function of frequency (2/λ where λ is thewavelength), and referring to FIGS. 7 and 8, one finds that PSD(R₀)peaks at

    ζ.sub.0 ≈nT.                                  (Eq. 1)

Similarly PSD (R_(k)), the power spectral density of the reflectionmagnitude at the oblique angle θ_(k) (FIG. 6), peaks at (FIGS. 7 and 8)##EQU3##

The ratio of the ζ_(k) to ζ₀ is given by ##EQU4## where θ_(k) and φ_(k)are the angle of incidence defined in air and inside the film 13respectively.

The index of refraction can be independently determined from Rζ_(k).##EQU5##

EXAMPLE 2

Reference is made to FIGS. 9 and 10 which pertain to Example 2. FIG. 9illustrates reflection amplitude as a function of wavelength ofpolyimide on silicon dioxide on silicon, and wherein the data wassimulated at both 0 and 70 degrees. FIG. 10 illustrates power spectraldensity of polyimide (8 μm) on silicon dioxide (2 μm) on silicon, andwherein the data was simulated at both 0 and 70 degrees.

The structure of the film 13 in this example is 8 μm polyimide on 2 μmsilicon dioxide on silicon substrate 14. The reflections at normalincidence and at an oblique angle of 70 degrees are shown in (FIG. 9).The power spectral density is shown in (FIG. 10). FIG. 10 clearly showsa significant shift in both the silicon dioxide (SiO₂) and polyimidepower spectral density peaks. These shifts in both peaks allow us tomeasure the index of refraction and thickness of both layerssimultaneously.

It is important to note that, in general, the present method 20 does notimpose any constraint on the value of wavelength dependence of index ofrefraction or extinction coefficient of the measured layer. The presentapparatus 10 and method 20 allows for index determination with highaccuracy (0.00001). The optical constants are modeled 26 using thedispersion formula, which accurately models the optical constants in themeasured wavelength range. The present method 20 can also be used withpolarized and random reflection and transmission data. Measurements atmultiple angles can be used for consistency check and higher accuracy.The present method 20 is not sensitive to errors in reflectionmagnitude.

Thus, nondestructive optical techniques for simultaneously measuringoptical constants and thicknesses of single and multilayer films hasbeen disclosed. It is to be understood that the above-describedembodiments are merely illustrative of some of the many specificembodiments that represent applications of the principles of the presentinvention. Clearly, numerous and other arrangements can be readilydevised by those skilled in the art without departing from the scope ofthe invention.

What is claimed is:
 1. A method for simultaneously determining opticalconstants and thicknesses of single and multilayer films, said methodcomprising the steps of:measuring predetermined spectra produced by thefilm at normal and oblique angles of incidence; computing power spectraldensity functions of the spectra at the normal and oblique angles;determining locations of statistically significant peaks of therespective power spectral density functions; modeling unknown parametersof the film to produce simulated values for the locations of thestatistically significant peaks; and comparing the measured andsimulated values of the locations of the statistically significant peaksto determine the thickness of the film or optical constants of the film.2. The method of claim 1 further comprising the steps of:computing aratio of the determined locations of the peaks; modeling unknownparameters of the film to produce simulated values for the ratio of thesimulated locations of the peaks; and comparing the measured andsimulated values of the locations of the statistically significant peaksand the ratios of the locations of the statistically significant peaksto determine the thickness of the film or optical constants of the film.3. The method of claim 1 wherein the comparing step comprisesiteratively comparing the measured and simulated values using an errorfunction that is minimized.
 4. The method of claim 3 wherein thecomparing step comprises the step of:changing the thicknesses andoptical constants of the simulated values and iteratively comparing themeasured and iterated simulated values until the error function isminimized.
 5. The method of claim 1 wherein:the determining step furthercomprises determining the magnitudes of the respective power spectraldensity functions; the modeling step further comprises modeling themagnitudes of the respective power spectral density functions; and thecomparing step also uses the measured and simulated values of themagnitudes of the power spectral density functions to determine thethickness of the film or optical constants of the film.
 6. The method ofclaim 4 wherein:the determining step further comprises determining therespective power spectral density functions; the modeling step furthercomprises modeling the respective power spectral density functions; andthe comparing step also uses the measured and simulated power spectraldensity functions to determine the thickness of the film or opticalconstants of the film.
 7. The method of claim 1 wherein the modelingstep comprises modeling the unknown parameters of the film using adispersion formula that describes the dispersion in the opticalconstants of the film.
 8. The method of claim 1 wherein the comparingstep comprises nonlinear global optimization algorithms that minimize amerit function of the measured and simulated data.
 9. The method ofclaim 5 wherein the predetermined spectra comprises spectra from thegroup including reflection spectra, transmission spectra, polarizedreflection spectra, polarized transmission spectra and ellipsometricspectra.
 10. Apparatus for simultaneously determining optical constantsand thicknesses of single and multilayer films, said apparatuscomprising:light sources and detectors for measuring predeterminedspectra produced by the film at normal and oblique angles of incidence;a computer for computing power spectral density functions of the spectraat the normal and oblique angles, for determining locations ofstatistically significant peaks of the respective power spectral densityfunctions, for computing a ratio of the determined locations of thepeaks, for modeling unknown parameters of the film to produce simulatedvalues for the locations of the statistically significant peaks and theratio of the simulated locations of the peaks, and for comparing themeasured and simulated values of the locations of the statisticallysignificant peaks and the ratios of the locations of the statisticallysignificant peaks to determine the thickness of the film or opticalconstants of the film.
 11. The apparatus of claim 10 wherein thecomputer:computes a ratio of the determined locations of the peaks;models unknown parameters of the film to produce simulated values forthe ratio of the simulated locations of the peaks; and compares themeasured and simulated values of the locations of the statisticallysignificant peaks and the ratios of the locations of the statisticallysignificant peaks to determine the thickness of the film or opticalconstants of the film.
 12. The apparatus of claim 10 wherein thecomparing performed by the computer comprises iteratively comparing themeasured and simulated values using an error function that is minimized.13. The apparatus of claim 12 wherein the comparing performed by thecomputer comprises changing the thicknesses and optical constants of thesimulated values and iteratively comparing the measured and iteratedsimulated values until the error function is minimized.
 14. Theapparatus of claim 10 wherein the computer:determines the magnitudes ofthe respective power spectral density functions; models the magnitudesof the respective power spectral density functions; and compares themeasured and simulated values of the magnitudes of the power spectraldensity functions to determine the thickness of the film or opticalconstants of the film.
 15. The apparatus of claim 14 wherein thecomputer:determines the respective power spectral density functions;models the respective power spectral density functions; and compares themeasured and simulated power spectral density functions to determine thethickness of the film or optical constants of the film.
 16. Theapparatus of claim 10 wherein the the computer models the unknownparameters of the film using a dispersion formula that describes thedispersion in the optical constants of the film.
 17. The apparatus ofclaim 10 wherein the predetermined spectra comprise a selected spectrafrom the group consisting of reflection spectra, transmission spectra,polarized reflection spectra, and polarized transmission spectra.